Question

The value $\underset{x\to {\tan }^{-1}3}{lim}\frac{{\tan }^{6}x-2{\tan }^{5}x-3{\tan }^{4}x}{{\tan }^{2}x-4\tan x+3}$ is

Answer 1

$\underset{x\to {\tan }^{-1}3}{lim}{\tan }^{4}x\frac{{\tan }^{2}x-2\tan x-3}{{\tan }^{2}x-4\tan x+3}$
$=\underset{x\to {\tan }^{-1}3}{lim}{\tan }^{4}x\frac{(\tan x-3)(\tan x+1)}{(\tan x-3)(\tan x-1)}$
$=\underset{x\to {\tan }^{-1}3}{lim}{\tan }^{4}x\frac{(\tan x+1)}{(\tan x-1)}=3^4\left(\frac{3+1}{3-1}\right)=162$